if i was in their shoes and had the choice between -

1. no passport, can’t leave anywhere in a normal way

2. a passport that is incorrect, can’t move comfortably because details that don’t match can and will trigger alarms and put eyes on you

3. a passport from a place that isn’t ideal, but matches my details, and allows me to travel without drawing attention to myself

i’d pick 3 as well regardless if i liked my country of origin or not